On the use of biorthogonality relations in the solution of some boundary value problems for the biharmonic equation

Shankar, PN (2004) On the use of biorthogonality relations in the solution of some boundary value problems for the biharmonic equation. Current Science, 85 (7). pp. 975-979.

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Abstract

Let a function amp;psi;(x,y), biharmonic in the semi-infinite strip {-1/2amp;lt;xamp;lt;1/2,yamp;lt;0}, be such that the function and its normal derivative vanish on the side walls x=amp;plusmn;1/2. We consider the problem of determining this function when we are given amp;psi;(x,0) and amp;nabla;lt;supgt;2lt;/supgt;amp;psi;(x,0) on the short edge y=0. First, we give a direct method of obtaining a biorthogonality relation among the eigenfunctions and then give a formal solution of the boundary value problem using this relation. Next we show that if we attempt to use this solution using a finite number of terms of the series, it is inferior to a solution where the expansion coefficients are calculating using a least squares procedure. This is a surprising result considering that for Fourier series, the Fourier coefficients are always optimal